Probability measure-valued polynomial diffusions

Sara Svaluto-Ferro,Martin Larsson,Christa Cuchiero

doi:10.1214/19-ejp290

Abstract

We introduce a class of probability measure-valued diffusions, coined polynomial, of which the well-known Fleming–Viot process is a particular example. The defining property of finite dimensional polynomial processes considered in [8, 21] is transferred to this infinite dimensional setting. This leads to a representation of conditional marginal moments via a finite dimensional linear PDE, whose spatial dimension corresponds to the degree of the moment. As a result, the tractability of finite dimensional polynomial processes are preserved in this setting. We also obtain a representation of the corresponding extended generators, and prove well-posedness of the associated martingale problems. In particular, uniqueness is obtained from the duality relationship with the PDEs mentioned above.

Highlights

We introduce a class of probability measure-valued diffusions, coined polynomial, of which the well-known Fleming–Viot process is a particular example
In this paper we develop probability measure-valued versions of a class of processes known as polynomial diffusions, which have – due to their inherent tractability – broad applications in population genetics, interacting particle systems, and finance; see e.g. [14, 33, 19]
A remarkably large class of processes is characterized via the polynomial property; in this paper, we find necessary and sufficient conditions for measure valued diffusions to be polynomial and to take values in the space of probability measures;

Summary

Introduction

In this paper we develop probability measure-valued versions of a class of processes known as polynomial diffusions, which have – due to their inherent tractability – broad applications in population genetics, interacting particle systems, and finance; see e.g. [14, 33, 19]. We work with martingale problems, but rather than using approximations by finite particle systems, we obtain existence directly via the positive maximum principle We obtain a full characterization of probability measure-valued polynomial diffusions whose generator L has a sufficiently large domain This yields extensions of the so-called Fleming–Viot process with weighted sampling discussed in [11, Section 5.7.8], where the samplingreplacement rate is allowed to depend on the type. N is a parameter that is chosen based on accuracy requirements, while d is an input to the problem This illustrates how probability measure-valued polynomial diffusions can enhance tractability in high-dimensional models.

Notation and basic definitions

Polynomials of measure arguments

Continuity and smoothness of polynomials

Polynomials with regular coefficients

Optimality conditions

Polynomial operators

Existence and well-posedness

Finite underlying space

Conditional laws of jump-diffusions are polynomial